Triangle Lectures in Combinatorics
Fall 2026 - 29th edition

September 26, 2026 at UNC Chapel Hill

Organizing Committee:  Laura Colmenarejo (NC State), Jaehyuk Kwon (UNC Chapel Hill), Ali Mohammad Nezhad (UNC Chapel Hill), Gabor Pataki (UNC Chapel Hill), and Fan Wei (Duke University)

Location

Chapman Hall, Room 0125 

Registration and Funding

Please, fill out this Google Form, to register for the conference and to request funding. Registration is free and funding is provided by NSF. If you have any questions, feel free to email any of the organizers.

We are asking that participants pre-register, if possible, as it is very helpful for planning our coffee breaks and obtaining funding to support these events. 

Poster session

Please, fill out this Google Form if you are interested in presenting a poster during the conference. The deadline is September 14, 2026.

Note that only junior researchers (students, postdocs, and tenure-track faculty) will present during the conference. Senior faculty should encourage their students and more junior colleagues to apply. For students, the poster they present must be on joint research with their advisor or a more senior researcher, and their name must appear as one of the authors. 

Parking Information: Please register by Thursday, Sept 24 at noon, if you would like a guaranteed and free parking spot.

Titles and Abstracts

Amir Ali Ahmadi:  Walking on Perfect Graphs with Sums of Squares in Mind

A graph is perfect if every induced subgraph has chromatic number equal to its clique number. In the first part of this talk, we give an algebraic characterization of this combinatorial property. We show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sums of squares. As a byproduct, we obtain several families of nonnegative polynomials that are not sums of squares through graph-theoretic constructions. In the second part of the talk, we present a polynomial-time algorithm for optimally coloring perfect graphs that is based entirely on graph-theoretic operations. At its core, the algorithm decides whether a perfect graph contains a clique of a given size by iteratively counting certain weighted walks.

Joint work with Cemil Dibek (first part) and with Pravesh Kothari and Yukai Tang (second part).

Saugata Basu:  Solving polynomial inequalities over spaces of convex sets and applications

Optimizing functions over convex sets is traditional. In this talk I will discuss optimization over spaces of convex sets. We develop a symbolic elimination theory for finite systems of recursive containment inequalities whose unknowns are convex subsets of a finite-dimensional real vector space.  Their right-hand sides are built from variables and parameters using convex combinations, finite unions, and a positive geometric join encoding strict convex combinations.  We prove that every parameter assignment has a unique smallest solution and give a finite, Gaussian-elimination-type procedure that eliminates the unknowns while preserving this solution.

The construction also gives a general closure principle: any class of sets closed under the relevant convex-geometric operations is preserved by the smallest-solution operator, effectively whenever those operations are effective.  In particular, finite unions of hemihedra (bounded convex semi-linear sets, equivalently convex finite unions of relative interiors of polytopes) lead to quantifier-free semi-linear descriptions.

The main application which links this elimination procedure to a problem in cryptopgraphy is to the theory of lamination hulls. Lamination hull operators are generalizations of the usual convex hull operator,  but they usually do not have a finite Carathéodory number. We prove that for an important family of such operators  (which arises in cryptographic and other applications), the lamination hull of any finite set of points is always semi-algebraic. This is nontrivial because the defining iterates need not stabilize and the resulting hull need not be semi-linear.

(Joint work with Hamidreza  A.  Khorasgani, Hemanta K. Maji and Hai H. Nguyen)

Evita Nestoridi: Limit Profiles for Reversible Markov Chains

A central question in Markov chain mixing is the occurrence of cutoff, a phenomenon according to which a Markov chain converges abruptly to its stationary measure. The focus of this talk is the limit profile of a Markov chain that exhibits cutoff, which captures the exact shape of the distance of the Markov chain from stationarity. We will discuss techniques for determining the limit profile and establishing its continuity properties under appropriate conditions.

Huy Pham: Random graphs: Typical and extreme behaviors 

Since their inception, random graphs have been a central topic in probability and combinatorics, offering a remarkably rich landscape for studying how complex structures emerge from randomness. The central question is deceptively simple: What is the probability that random graphs contain certain structures? Studying different regimes of such containment probability leads to the heart of different subjects with rich understanding and fundamental phenomena. 

On one hand, in the typical regime, the study of thresholds concerns the behavior of the containment probability where it transitions around 1/2, as such describing the typical behavior of random graphs. On the other hand, in the extreme tail where the containment probability is very close to 1, understanding the asymptotics of the tail probability is an important topic in the study of large deviations. 

In this talk, I will discuss recent progress on both sides of this spectrum. I will describe the fundamental forces governing containment probabilities in the typical and extreme regimes, and highlight surprising connections to some of the cornerstone ideas and developments in modern combinatorics and probability.